All correct. But there's still a decision made to act. Pick up the knife, jump off the bridge, hang the belt from the rafters, pull the trigger, Etc. You are choosing to depart the mortal. It's deliberate.
That bothers people. They want a tidier solution. Deep feeling of despair from depression, etc.
Someone who had to get the belt from the pants, move the chair somewhere, loop the belt and latch it, move under it, and sit... Every one of those actions is a choice... AFTER he already tried cutting his wrists. He made the decision over and over and over.
Not what people want to hear. I know. But I won't disrespect his decision trying to find a simple answer for it. What happened in that room was a fairly long slow deliberate process.
What bothers people is that they have thought similar thoughts and wonder what kept them from it and will that always be there. What bothers people is someone so selfishly forced them to face their own mortality and mental demons. People don't like that **** so they ignore it most of the time. A depressive can't ignore it. Suicide is a deliberate act of that there is no argument, however it is a deliberate act that alters things in timing only, not the actual result.
Think about what it would have been like to be Robin Williams with enough voices popping into his head he was one bad link from being schizophrenic. Dealing with a mind like that can be extremely tiring and taxing. When you read a story to your kid and they ask you not to do the voices and be yourself, and realizing the voices are yourself, that's tough.